Algebras of approximation sequences: Fredholm theory in fractal

Algebras of approximation sequences: Fredholm theory in fractal

STUDIA MATHEMATICA 150 (1) (2002)
Algebras of approximation sequences:
Fredholm theory in fractal algebras
by
Steffen Roch (Darmstadt)
Abstract. The present paper is a continuation of [5, 7] where a Fredholm theory
for approximation sequences is proposed and some of its properties and consequences are
studied. Here this theory is specified to the class of fractal approximation methods. The
main result is a formula for the so-called α-number of an approximation sequence (A n )
which is the analogue of the kernel dimension of a Fredholm operator.
1. Introduction. Let H be a complex Hilbert space, L(H) the C ∗ algebra of bounded linear operators on H, and K(H) the ideal of compact
linear operators on H. Further let (Pn ) be a sequence of orthogonal projections on H which converges strongly to the identity operator I on H:
s-limPn = I ⇔ Pn x → x for all x ∈ H,
and assume that dim Im Pn = n. Thus, Im Pn and L(Im Pn ) can be identified
with the linear space Cn and with the algebra Cn×n = L(Cn ), respectively.
The n × n identity matrix will be denoted by In .
Let A ∈ L(H). An approximation method for A is a sequence (An ) of
matrices An ∈ Cn×n such that An Pn → A and A∗n Pn → A∗ strongly as
n → ∞. This method converges if the equations
An x(n) = Pn y
have unique solutions x(n) ∈ Im Pn for all sufficiently large n and all right
hand sides y ∈ H, and if these solutions converge in the norm of H to a
solution of the equation
Ax = y.
By the Banach–Steinhaus theorem, the method (An ) for A is applicable if
and only if the sequence (An ) is stable in the sense that the matrices An are
invertible for all sufficiently large n and
sup kA−1
n k < ∞.
2000 Mathematics Subject Classification: Primary 46N40; Secondary 65R20.
[53]
54
S. Roch
For another characterization of stability, introduce the set F of all bounded
sequences (An ) of matrices An ∈ Cn×n . Provided with elementwise operations and the supremum norm, this set becomes a C ∗ -algebra with identity
element (In ), and the set G of all sequences (An ) ∈ F with lim kAn k = 0
forms a closed two-sided ideal in F. A Neumann series argument shows that
the sequence (An ) ∈ F is stable if and only if its coset (An ) + G is invertible
in the quotient algebra F/G.
The role of the algebra F/G in numerical analysis can be compared with
the role of the algebra L(H) in operator theory. In this sense, the stable
sequences in F correspond to the invertible operators on H. For operators,
there is a useful generalization of the notion of invertibility: one considers
operators A ∈ L(H) which are almost invertible in the sense that both their
kernel Ker A and their cokernel Coker A := H/Im A have finite dimension.
Operators with this property are called Fredholm operators, and the quantity
ind A := dim Ker A − dim Coker A is referred to as the index of A.
In [7, 5] an analogous notion of a Fredholm approximation sequence is
introduced and studied. The kernel dimension condition for a Fredholm
operator is replaced by the following condition for a Fredholm sequence
(An ) ∈ F: There is an ε > 0 and a fixed number k such that at most k
singular values of An are less than ε for all sufficiently large n. The smallest
possible number k with this property is the analogue of the kernel dimension for the Fredholm sequence (An ), and we call it the α-number of that
sequence. One can also define the index of a Fredholm sequence in a natural
way, but it turns out that this index is always zero. This is a consequence
of the fact that the approximation operators An which we consider act on
finite-dimensional spaces. Exact definitions will be given in Section 2.
So the most important quantity related to a Fredholm sequence seems
to be its α-number. In [7], we derived a simple and practicable formula to
compute this number for large classes of concrete approximation sequences.
(In fact, for all sequences which lie in a standard algebra in the terminology of [7]. This class includes, e.g., projection methods with spline ansatz
functions for singular integral operators or quadrature formula methods for
Mellin operators, to mention only a few.) It has been one goal in [5] to generalize this formula to the class of all fractal approximation methods. This
class has been introduced in [6, 4]. Roughly speaking, a sequence (An ) is
fractal (or self-similar) if (An ) can be completely reconstructed from each
of its infinite subsequences up to a sequence tending to zero in norm. The
precise definition is in Section 3. The class of fractal sequences includes all
sequences in standard algebras.
For the formula for the α-numbers derived in [7] one needs an additional
hypothesis which is satisfied for all sequences in standard algebras but not
for all fractal sequences. In the present paper, we will set forth a specification
Algebras of approximation sequences
55
of the general Fredholm theory (including a formula for the α-numbers) to
fractal approximation methods without any additional (and hard to check)
hypotheses (Section 4).
For applications of this Fredholm theory to the analysis of concrete approximation methods we refer to the already mentioned papers [7, 8, 5] and
to the textbook [2]. Nice applications to a special class of approximation
sequences (the finite section method for Toeplitz operators) are in [1].
2. Compact and Fredholm sequences. In this section we recall the
definition of a compact and a Fredholm approximation sequence from [5],
and we derive some characterizations of compact sequences which are basic
in what follows.
2.1. The ideal of compact sequences
Sequences of rank one matrices. Let K be the smallest closed ideal of F
which contains all sequences of matrices of rank ≤ 1. The product of such
a sequence with another sequence in F is again a sequence of matrices of
rank ≤ 1. Hence, the set K0 of all finite sums of sequences matrices of rank
≤ 1 forms an ideal (in general, non-closed) of F whose closure is just the
ideal K. Consequently, a sequence (An ) ∈ F belongs to K if and only if, for
every ε > 0, there is a sequence (Kn ) ∈ F such that
(1)
sup kAn − Kn k < ε and
sup dim Im Kn < ∞.
n
n
We refer to the elements of K as compact sequences. The role of the ideal K
of compact sequences in numerical analysis can be compared with the role
of the ideal of compact operators in operator theory.
Characterization of compact sequences via singular values. Given a sequence (Kn ) ∈ F, denote by
(n)
(2)
0 ≤ σ1
(n)
≤ σ2
≤ . . . ≤ σn(n)
the singular values of Kn , i.e. the non-negative square roots of the eigen(n)
(n)
values of Kn∗ Kn . Further set Σk := σn−k+1 for k = 1, . . . , n. Clearly,
(n)
(n)
kKn k = σn = Σ1 .
Theorem 2.1. A sequence (Kn ) ∈ F is compact if and only if
(3)
(n)
lim sup Σk
k→∞ n≥k
= 0.
Proof. Let (Kn ) ∈ F satisfy (3), and let (with unitary matrices Un , Vn )
(n)
Kn = Un diag(Σ1 , . . . , Σn(n) )Vn∗
56
S. Roch
be the singular value decomposition of Kn . For every

(n)
 Un diag(Σ1(n) , . . . , Σk−1
, 0, . . . , 0)Vn∗
(k)
Kn := 0

Kn
Then, for all k, n ≥ 1,
kKn − Kn(k) k
k, n ≥ 1, set
if 1 < k ≤ n,
if 1 = k ≤ n,
if n < k.

 kUn diag(0, . . . , 0, Σk(n) , . . . , Σn(n) )Vn∗ k = Σk(n)
=
 kKn k =
0
Hence,
(n)
Σ1
if 1 < k ≤ n,
if 1 = k ≤ n,
if n < k.
(n)
k(Kn )n≥1 − (Kn(k) )n≥1 kF = sup Σk ,
n≥k
which together with hypothesis (3) implies
(n)
lim k(Kn )n≥1 − (Kn(k) )n≥1 kF = lim sup Σk
k→∞
k→∞ n≥k
= 0.
(k)
Thus, the sequence (Kn ) is the limit as k → ∞ of the sequences (Kn )n≥1 .
(k)
(k)
Since dim Im Kn ≤ k−1, each (Kn )n≥1 lies in K0 , and so (Kn ) is compact.
(n)
Let now (Kn ) be a compact sequence. Then (supn≥k Σk )k≥1 is decreasing and bounded below (by zero) and, hence, convergent. Assume that its
limit is positive. Then there is a C > 0 such that
(n)
sup Σk
n≥k
≥C
for all k ≥ 1.
Hence, for each k ≥ 1, there is an nk ≥ k such that
(nk )
(4)
Σk
≥C
for all k ≥ 1.
Further, since (Kn ) is compact, there is a sequence (Rn ) ∈ F with
sup dim Im Rn < ∞
n
and
sup kKn∗ Kn − Rn∗ Rn k < C
n
(cf. (1)). Set r := supn dim Im Rn and choose k > r. Thus, in particular,
(5)
Let now
kKn∗k Knk − Rn∗ k Rnk k < C.
(nk )
Kn∗k Knk = Unk diag(Σ1
, . . . , Σn(nkk ) )Un∗k
with a unitary matrix Unk , and set
Pk := diag(1, . . . , 1, 0, . . . , 0)
with k ones and nk − k zeros. Then, obviously, the matrix
(nk )
Pk Un∗k Kn∗k Knk Unk Pk = diag(Σ1
(nk )
, . . . , Σk
, 0, . . . , 0)
Algebras of approximation sequences
is invertible on Im Pk , and
k(Pk Un∗k Kn∗k Knk Unk Pk |Im Pk )−1 k
= max
1
(n )
Σ1 k
57
,...,
whence
1
(n )
Σk k
(nk )
k(Pk Un∗k Kn∗k Knk Unk Pk |Im Pk )−1 k−1 = Σk
due to (4). Since, by (5),
=
1
(n )
Σk k
,
≥C
kPk Un∗k Kn∗k Knk Unk Pk − Pk Un∗k Rn∗ k Rnk Unk Pk k < C,
a Neumann series argument yields the invertibility of Pk Un∗k Rn∗ k Rnk Unk Pk ,
considered as an operator on Im Pk . This implies that
dim Im Rn∗ k Rnk ≥ dim Im Pk = k,
(n)
which contradicts our assumption. Hence, the sequence (supn≥k Σk )k≥1
cannot have a positive limit, i.e. (Kn ) satisfies (3).
(n)
Corollary 2.2. Let (Kn ) ∈ K. Then limn→∞ σk
every k.
exists and is 0 for
(n)
Proof. Let ε > 0. By Theorem 2.1, there is a k0 such that supn≥k0 Σk0
< ε. Then, for all n ≥ n0 := k0 + k − 1,
(n)
σk
(n)
(n)
(n)
= Σn−k+1 ≤ Σk0 ≤ sup Σk0 < ε.
n≥k0
Further characterizations of the ideal K. We proceed with two further
characterizations of the ideal K which are partially based on the results of
the preceding subsection. One aim of these characterizations is to show that
the above definition of compact sequences is in some sense the only possible
one. Indeed, suppose for a moment that we have no idea of what a compact
sequence might be and that we would like to introduce an ideal of F which
corresponds to the ideal of compact operators on a Hilbert space. The concrete approximation methods considered and mentioned in [6] suggest that
the constant sequence (P1 ) where P1 is the diagonal matrix diag(1, 0, . . . , 0)
should be considered as compact in any case. Thus, a minimal candidate
for the desired ideal of compact sequences is the smallest closed ideal of F
which contains the sequence (P1 ). There is also a natural largest candidate:
the ideal of all sequences (Kn ) such that W (Kn ) is a compact operator for
every irreducible representation W of F. The characterizations below show
that actually both candidates coincide with the ideal K defined before.
Theorem 2.3. K is the smallest closed ideal of F which contains the
constant sequence (P1 ).
Proof. Let, for a moment, K0 stand for the smallest closed ideal of F
which contains (P1 ). It is evident that K0 ⊆ K. For the reverse inclusion, we
58
S. Roch
have to verify that every sequence (Kn ) ∈ F of rank ≤ 1 matrices belongs
to K0 .
Indeed, for such a sequence define
kKn∗ Kn k−1 Kn∗ Kn if Kn 6= 0,
0
Kn :=
P1
if Kn = 0.
Every matrix Kn0 is an orthogonal projection of rank one. Hence, there are
unitary matrices Un ∈ Cn×n such that
Kn0 = Un∗ diag(1, 0, . . . , 0)Un = Un∗ P1 Un .
The sequence (Un ) is bounded and belongs therefore to F. Hence, (Kn0 ) =
(Un∗ )(P1 )(Un ) belongs to K0 , and so does (Kn ) because of
1 if Kn 6= 0,
(Kn ) = (λn )(Kn )(Kn0 ) with λn =
0 if Kn = 0.
Theorem 2.4. (a) If (Kn ) ∈ F is a sequence of rank ≤ 1 matrices and
(H, π) is an irreducible representation of F, then π(Kn ) is an operator with
rank ≤ 1.
(b) A sequence (Kn ) ∈ F belongs to the ideal K if and only if π(Kn ) is
compact for every irreducible representation (H, π) of F.
Proof. Assertion (a) is proved in [5] (Proposition 2 and Theorem 3). For
a proof of (b), let K0 stand for the set of all sequences (Kn ) in F such that
π(Kn ) is compact for every irreducible representation π of F. The inclusion
K ⊆ K0 is an immediate consequence of (a). For the proof of the reverse
inclusion, we will show that, for every sequence (Kn ) ∈ F \ K, there is an
irreducible representation π of F such that π(Kn ) is not compact.
Let (Kn ) ∈ F be a sequence which is not in K, and let Id(Kn ) be the
smallest closed ideal of F which contains the sequence (Kn ). Further, let
(n)
(n)
Λ1 ≥ . . . ≥ Λn ≥ 0 denote the eigenvalues of Kn∗ Kn , and let Un be a
unitary matrix such that
(n)
Un∗ Kn∗ Kn Un = diag(Λ1 , . . . , Λ(n)
n ).
(n)
Since (Kn ) 6∈ K, Theorem 2.1 implies that limk→∞ supn≥k Λk =
6 0. Thus,
as we checked in the proof of Theorem 2.1, there is a C > 0 as well as a
sequence (nk )k≥1 with nk ≥ k such that
(6)
(nk )
Λk
≥C
for all k.
The sequence (nk ) can be chosen strictly increasing. Indeed, suppose that we
have already determined n1 < . . . < nk . Choose k 0 > nk and a corresponding
(n0 )
n0 ≥ k 0 such that Λk0 ≥ C, which is possible due to (6). Then nk+1 := n0
satisfies nk+1 = n0 ≥ nk +1 ≥ k +1, yielding the desired strict monotonicity.
Algebras of approximation sequences
59
So let (nk ) be a strictly increasing sequence satisfying (6). Set
(nk )
Ln := diag(1/Λ1
(nk )
, . . . , 1/Λk
, 0, . . . , 0)
if n = nk for some k, and Ln := 0 if n 6= nk . The sequence (Ln ) is bounded
(n )
because (Λj k )−1 ≤ 1/C for j = 1, . . . , k. Hence, the sequence (Dn ) with
0
if n 6= nk ,
∗ ∗
Dn := Ln Un Kn Kn Un =
diag(1, . . . , 1, 0, . . . , 0) if n = nk ,
with k ones and nk − k zeros belongs to the ideal Id(Kn ). Then all sequences
(7)
(Dn Pn APn Dn )
with A ∈ L(l2 )
lie in Id(Kn ), and we consider the smallest closed C ∗ -subalgebra B of Id(Kn )
which contains all sequences (7). Clearly, the strong limit s-limk→∞ Bnk
exists for every sequence (Bn ) ∈ B and, in particular,
s-lim Dnk Pnk APnk Dnk = A
k→∞
for all sequences of the form (7). The mapping
W : B → L(l2 ),
(Bn ) 7→ s-lim Bnk ,
k→∞
is an irreducible representation of B (all compact operators lie in the range of
W ) which maps (Dn ) to the identity operator on l 2 . From Proposition 4.1.8
of [3] we know that there exists an irreducible representation π : F → L(H),
a closed subspace H1 of H, and a bijective isometry U : H1 → l2 such that
W (Bn ) = U π(Bn )|H1 U ∗
(8)
for all (Bn ) ∈ B.
The operator π(Kn ) cannot be compact. Indeed, suppose that it is. Then
the operators π(Rn ) are compact for all sequences (Rn ) ∈ Id(Kn ), hence so
are the operators π(Dn ). Then, by (8), W (Dn ) must be compact. This is
impossible since W (Dn ) is the identity operator on l 2 as we already observed.
Thus, (Kn ) 6∈ K0 , yielding the inclusion K0 ⊆ K.
2.2. Fredholm sequences. Corresponding to the ideal K we introduce an
appropriate class of Fredholm sequences by calling a sequence (An ) ∈ F
Fredholm if it is invertible modulo K (see [5]). The following properties of
Fredholm sequences are obvious.
•
•
•
•
•
Every stable sequence is Fredholm.
The adjoint of a Fredholm sequence is Fredholm.
The product of Fredholm sequences is Fredholm.
If (An ) is Fredholm and (Kn ) ∈ K, then (An + Kn ) is Fredholm.
The set of Fredholm sequences is open in F.
(n)
For another characterization of Fredholm sequences, let 0 ≤ σ1
(n)
σn denote the singular values of a sequence (An ) ∈ F.
≤ ... ≤
60
S. Roch
Theorem 2.5. Each of the following conditions is equivalent to the Fredholmness of a sequence (An ) ∈ F:
(a) There are sequences (Bn ) ∈ F and (Jn ) ∈ K with supn dim Im Jn
< ∞ such that
(9)
Bn A∗n An = In + Jn .
(b) There is a k such that
(10)
(n)
lim inf σk+1 > 0.
n→∞
This theorem suggests introducing the α-number α((An )) of a Fredholm sequence (An ) (corresponding to the kernel dimension of a Fredholm
operator) as the smallest number k for which (10) is true. Equivalently,
α((An )) is the smallest number for which there exists a sequence (Bn ) ∈
F as well as a sequence (Jn ) ∈ K such that Bn A∗n An = In + Jn and
lim supn→∞ dim Im Jn = α((An )). The index of a Fredholm sequence is the
quantity
ind((An )) := α((An )) − α((A∗n )).
Observe that, in the case at hand, this index is always zero. This is a consequence of the fact that the entries of the sequences under consideration are
finite-dimensional operators and, hence, the matrices A∗n An and An A∗n have
the same eigenvalues even with respect to their multiplicity. So the most
interesting quantity associated with a Fredholm sequence of matrices seems
to be its α-number. On the other hand, the vanishing of the index of (An )
has remarkable consequences as has been pointed out in [5].
Let us still emphasize that, in general, the singular values of a compact
or a Fredholm sequence do not exhibit the behaviour one would expect from
the knowledge about the behaviour of the singular values of a compact or a
Fredholm operator. Indeed, if K is a compact operator on a Hilbert space,
then there are at most countably many singular values, and all non-zero
singular values are isolated points. Similarly, if A is a Fredholm operator on
a Hilbert space, then its range is closed and A is Moore–Penrose invertible.
The latter property is satisfied if and only if either A∗ A is invertible or if 0 is
an isolated point in the spectrum of A∗ A. Thus, one might expect that the
non-zero singular values of the coset (Kn )+G of a compact sequence (Kn ) or
the singular value 0 of the coset (An ) + G of a non-stable Fredholm sequence
(An ) are isolated. But this is not true as the following simple example shows.
Example. Let (an ) be an enumeration of the rational numbers in [0, 1],
and set
Kn := an Pn P1 Pn = diag(an , 0, . . . , 0),
An := Kn + Pn (I − P1 )Pn = diag(an , 1, . . . , 1).
Algebras of approximation sequences
61
The sequence (Kn ) consists of rank ≤ 1 matrices; hence (Kn ) is a compact
and (An ) is a Fredholm sequence. But both the largest singular values of the
matrices Kn and the smallest singular values of the matrices An lie dense
in [0, 1]. Thus, the spectrum of both cosets (Kn∗ Kn ) + G and (A∗n An ) + G is
[0, 1].
We will see in the following sections that compact and Fredholm sequences in fractal subalgebras show a behaviour of the singular values which
is in perfect accordance with the behaviour of the singular values of compact
and Fredholm operators.
3. Compact sequences in fractal algebras. In this section, we consider compact sequences (Kn ) in fractal algebras. In particular, the announced behaviour of the singular values of the matrices Kn will be established. We start by recalling the definition of a fractal algebra and of
some properties of fractal algebras from [7, 5].
Given a strongly increasing sequence η : N → N, let Fη be the C ∗ -algebra
of all bounded sequences (An ) with An ∈ Cη(n)×η(n) , and write Gη for the
ideal of all sequences (An ) ∈ Fη which tend to zero in norm. Further, let
Rη stand for the restriction mapping Rη : F → Fη , (an ) 7→ (aη(n) ). This
mapping is a ∗ -homomorphism from F onto Fη which moreover maps G onto
Gη . Further, given a C ∗ -subalgebra A of F, let Aη denote the image of A
under Rη which is a C ∗ -algebra again.
Definition 3.1. Let A be a C ∗ -subalgebra of the algebra F.
(a) A ∗ -homomorphism W : A → B of A into a C ∗ -algebra B is fractal
if, for every strongly increasing sequence η, there is a ∗ -homomorphism
Wη : Aη → B such that W = Wη Rη .
(b) The algebra A is fractal if the canonical homomorphism π : A →
A/(A ∩ G), (An ) 7→ (An ) + (A ∩ G), is fractal.
(c) A sequence (An ) ∈ F is fractal if the smallest C ∗ -subalgebra of F
which contains (An ) is fractal.
Thus, given a subsequence (Aη(n) ) of a sequence (An ) which belongs to
a fractal algebra A, it is possible to reconstruct the original sequence (An )
from this subsequence modulo sequences in A ∩ G. This assumption is very
natural for sequences arising from discretization procedures, and it is indeed
satisfied for a lot of concrete approximation sequences; see e.g. [6]. On the
other hand, the algebra F of all bounded sequences fails to be fractal.
Here are a few properties of fractal sequences proved in [5].
Theorem 3.2. A C ∗ -subalgebra A of F is fractal if and only if , for
every (An ) ∈ A and every strongly increasing sequence η,
Rη (An ) ∈ Gη ⇒ (An ) ∈ A ∩ G.
62
S. Roch
Theorem 3.3. A sequence (An ) ∈ F of self-adjoint matrices is fractal
if and only if lim inf σ(An ) = lim sup σ(An ) where σ(An ) stands for the
spectrum of An .
If the limes superior lim sup Mn and the limes inferior lim inf Mn of a
sequence (Mn ) of compact subsets of the complex plane coincide, then the
sequence (Mn ) converges with respect to the Hausdorff distance, and we
write lim Mn for the limit in the Hausdorff sense.
Proposition 3.4. If (An ) ∈ F is a fractal sequence, then lim kAn k
exists and it is equal to the norm of the coset (An ) + G in the quotient
algebra F/G.
3.1. Singular values of fractal compact sequences. Let (Kn ) ∈ F be a
(n)
fractal sequence (not necessarily compact for the moment), and let Σ1 ≥
(n)
. . . ≥ Σn ≥ 0 denote the singular values of Kn . Further we write σ2 (A) for
the set of singular values of an operator A.
(n)
Lemma 3.5. The sequence (Σ1 ) of the largest singular values of Kn
converges.
Proof. The assertion is an immediate consequence of Proposition 3.4
(n)
and of the identity Σ1 = kKn k. Another short proof based on Theorem
(n)
3.3 runs as follows. Assume that (Σ1 ) has two limiting points, say α and
(n )
β with α > β. Then β is the limit of a subsequence (Σ1 r )r≥1 . The point
(n )
α cannot belong to lim supr→∞ σ2 (Knr ) (the Σ1 r are the largest singular
values of Knr , and they converge to β which is less than α). Hence,
α ∈ lim sup σ2 (Kn ) \ lim inf σ2 (Kn ),
in contradiction to Theorem 3.3.
(n)
Observe that the sequence (Σ2 ) need not converge even if (Kn ) is a
fractal sequence, as shown by the example
diag(1, 0, 0, . . . , 0) if n is even,
Kn :=
diag(1, 1, 0, . . . , 0) if n is odd.
(n)
But, as in the second proof of Lemma 3.5, one can easily verify that (Σ2 )
can have at most two limiting points, at most one of which can be different
(n)
from lim Σ1 . More generally, the following is true, where we write Πk for
(n)
the set of all partial limits of the sequence (Σk )n≥k .
Proposition 3.6. If (Kn ) ∈ F is fractal , then Πk+1 \ Πk contains at
most one element.
Algebras of approximation sequences
63
Proof. We start by showing the identity
(n)
(11)
(n)
Π1 ∪ . . . ∪ Πk = lim sup{Σ1 , . . . , Σk }
n→∞
for every k.
(n)
(n)
The inclusion ⊆ is evident. Conversely, if λ ∈ lim supn→∞ {Σ1 , . . . , Σk },
then there are a subsequence (nr ) of N and numbers kr in {1, . . . , k} such
that
(n )
λ = lim Σkr r .
r→∞
Since the kr can take only finitely many values, there is a k0 between 1 and
k such that
(n )
λ = lim Σk0 r .
r→∞
Hence, λ ∈ Πk0 , which shows (11).
Now we proceed as in the second proof of Lemma 3.5. Assume that the
(n)
sequence (Σk+1 )n has two limiting points α and β with α > β, both not
(n )
r
belonging to Πk . Choose a subsequence (Σk+1
)r≥1 which converges to β as
r → ∞. Then α cannot belong to lim supr→∞ σ2 (Knr ). Indeed, since α 6∈ Πk
by assumption, α cannot belong to
(n)
(n)
Π1 ∪ . . . ∪ Πk = lim sup{Σ1 , . . . , Σk }
n→∞
due to monotony reasons. But the (k + 1) th singular values of Knr converge
to β < α. Hence,
α ∈ lim sup σ2 (Kn ) \ lim inf σ2 (Kn ),
in contradiction to Theorem 3.3.
(n)
Corollary 3.7. If (Kn ) ∈ F is fractal , then lim supn→∞ {Σ1 , . . .
(n)
. . . , Σk } contains at most k elements.
Now we can show that, with respect to the singular values, fractal compact sequences behave as compact operators on Hilbert space.
Corollary 3.8. Let (Kn ) ∈ K be a fractal sequence and ε > 0. Then
there are only finitely many points in lim σ2 (Kn ) with absolute value larger
than ε.
(n)
Proof. By Theorem 2.1, limk→∞ supn≥k Σk
such that
(n)
supn≥k0 Σk0
= 0. Thus, there is a k0
≤ ε, whence
(n)
lim sup{Σk0 , . . . , Σn(n) } ⊆ [0, ε].
n→∞
Since
(n)
(n)
(n)
lim σ2 (Kn ) = lim sup{Σ1 , . . . , Σk0 −1 } ∪ lim sup{Σk0 , . . . , Σn(n) },
n→∞
n→∞
n→∞
64
S. Roch
it is obvious that the points in lim σ2 (Kn ) with absolute value larger than ε
lie in
(n)
(n)
lim sup{Σ1 , . . . , Σk0 −1 }.
n→∞
This set is finite by Corollary 3.7.
Corollary 3.9. Let (Kn ) ∈ K be fractal. Then 0 ∈ lim σ2 (Kn ), the set
lim σ2 (Kn ) is at most countable, and 0 is its only limiting point.
3.2. The ideal (A ∩ K)/G. Throughout what follows let A be a unital
fractal C ∗ -subalgebra of F which contains the ideal G. We will now apply
the results of the preceding section to analyse the ideal A ∩ K of compact
sequences in A.
Generators of (A ∩ K)/G. Our first goal is to show that the ideal A ∩ K
is generated by its projections. Later on we will see that it is even generated
by its minimal projections.
Theorem 3.10. The ideal A ∩ K is generated (as a C ∗ -algebra or as an
ideal of A) by its projections, i.e. by the sequences (Qn ) ∈ A ∩ K where each
matrix Qn is a self-adjoint idempotent.
Proof. Every sequence in A ∩ K is a linear combination of four nonnegative sequences in A ∩ K, and every non-negative sequence in A ∩ K is of
the form (Kn∗ Kn ) with a sequence (Kn ) ∈ A ∩ K. The assertion will follow
once we show that, for every sequence (Kn ) ∈ A ∩ K, the sequence (Kn∗ Kn )
can be approximated in the norm of F by linear combinations of projections
in A ∩ K.
By Corollary 3.9, there are (finitely or countably many) numbers λ1 >
λ2 > . . . > 0 such that
σF ((Kn∗ Kn )) = {λ1 , λ2 , . . .} ∪ {0} =: Λ.
The smallest C ∗ -subalgebra K0 of F which contains the sequences (Kn∗ Kn )
and (In ) is ∗ -isometric to C(Λ), and the Gelfand transform maps (Kn∗ Kn ) to
the identity mapping on Λ. For every λk , let Pk be the sequence in K0 = C(Λ)
whose Gelfand transform is 1 at λk and 0 on Λ \ {λk } (this function is
obviously continuous on Λ). Each Pk is a projection in K0 ⊆ A ∩ K, and
r
n
X
∗
λk+1 if Λ has at least k + 2 elements,
(K
K
)
−
λ
P
=
n n
i i
0
if Λ has at most k + 1 elements.
F
i=1
Since λk → 0 as k → ∞, this implies the assertion.
Due to Theorem 3.3, one of the following possibilities is satisfied for every
fractal sequence (Qn ) of projections: either σ(Qn ) = {1}, or σ(Qn ) = {0},
or σ(Qn ) = {0, 1}, for all sufficiently large n. In the first case we have
Qn = In for all sufficiently large n, and the sequence (Qn ) cannot belong
Algebras of approximation sequences
65
to K (Corollary 2.2). In the second case, Qn = 0 for all large n, and the
sequence (Qn ) lies in G.
We call a sequence (Qn ) ∈ F of projections essential if 1 ∈ σ(Qn ) for
all n.
Proposition 3.11. The ideal A ∩ K is generated (as an ideal in A) by
its essential projections.
Proof. For a moment, let K0 stand for the smallest closed ideal of A
which contains all essential projections from A ∩ K. Evidently, K 0 ⊆ A ∩ K.
For the reverse inclusion, we have to show that K 0 contains all projections
in A ∩ K (Theorem 3.10). For this goal, we first check that
(12)
G ⊆ K0 .
Let r ∈ N and (Qn ) ∈ A ∩ K be an essential sequence of projections. The
sequence (0, . . . , 0, Ir , 0, . . .) with Ir standing at the rth place lies in A; hence,
the sequence (0, . . . , 0, Qr , 0, . . .) belongs to K0 . Since Qr 6= 0, the smallest
closed ideal of Cr×r which contains Qr is equal to Cr×r , which shows that
K0 contains all sequences (0, . . . , 0, Gr , 0, . . .) with Gr ∈ Cr×r . Consequently,
all sequences (G1 , G2 , . . . , Gk , 0, 0, . . .) belong to K0 . Since G is the closure
of the set of all sequences of this kind, (12) holds.
Let now (Rn ) ∈ A ∩ K be a projection. By the above classification, either
σ(Rn ) = {0} or σ(Rn ) = {0, 1}, for all sufficiently large n. In the first case,
(Rn ) ∈ G, whence (Rn ) ∈ K0 due to (12). In the second case, at most finitely
many of the matrices Rn are zero. We replace them by In and get a new
sequence (Rn0 ). This sequence is an essential projection and, hence, is in K 0
by definition, and the difference (Rn0 ) − (Rn ) is in G by construction and in
K0 due to (12).
Later, we will need the following simple consequence of the preceding
proposition.
Corollary 3.12. If q ∈ (A ∩ K)/G is a non-zero projection, then there
is an essential projection (Qn ) ∈ A ∩ K such that (Qn ) + G = q.
We also say that q can be lifted to an essential projection. For the proof
note that every projection q ∈ F/G can be lifted to a projection (Rn ) ∈ F
(see, e.g., [7]). Since q 6= 0 and q ∈ K/G, we have (Rn ) 6∈ G and (Rn ) 6∈
(Pn ) + G. Thus, σ(Rn ) = {0, 1} for all sufficiently large n, and (Rn ) can be
made an essential projection by adding a sequence in G as in the proof of
Proposition 3.11.
An order relation for projections. In what follows we will have to compare projections in (A ∩ K)/G. The needed definitions will be given in a
more general context.
66
S. Roch
Let B be a C ∗ -algebra, and let q, r be projections (i.e. self-adjoint idempotents) in B. We say that r q if rqr = r. If r q and r 6= q, then we
write r ≺ q. Obviously, 0 q, and if B has an identity element e, then q e
for every projection q.
If rqr = r for projections q, r, then (i(qr −rq))3 = 0 as one easily verifies.
Since i(qr−rq) is self-adjoint, this observation implies that qr = rq whenever
rqr = r. With this commutativity, it is trivial to check that the relation is reflexive, symmetric and transitive, i.e. an order relation.
Lemma 3.13. Let q, r be projections in a C ∗ -algebra B such that r q.
Then q − r is also a projection, and q − r q.
Proof. We have already remarked that r q implies rqr = rq = qr = r.
Hence,
(q − r)2 = q 2 − qr − rq + r 2 = q − r
and
(q − r)q(q − r) = q 3 − q 2 r − rq 2 + rqr = q − qr − rq + r = q − r.
The first identity shows that q−r is a projection, the second that q−r q.
We call a projection p ∈ B minimal if, for every projection q ∈ B with
q p, q = 0 or q = p. Clearly, 0 is a minimal projection, the trivial minimal
projection. The quotient l ∞ /c0 is an example of a C ∗ -algebra without nontrivial minimal projections. Further, an element r ∈ B is said to be of rank
one if, for every b ∈ B, there is a β ∈ C such that rbr = βr.
Proposition 3.14. Let B be a C ∗ -algebra. Then every projection of
rank one is minimal.
Proof. Let p ∈ B be a projection of rank one. If p = 0, then there is
nothing to prove. So let p 6= 0, and let q ∈ B be a projection with q p,
i.e. qpq = q. Since p is of rank one, there is a µ ∈ C such that pqp = µp. So
one has
q = q 2 = (qpq)2 = q(pqp)q = µqpq = µq,
whence q = 0 or µ = 1. If µ = 1, then pqp = p, i.e. p q. Due to the
symmetry of , this implies p = q. Thus, in any case, q = 0 or q = p, i.e. p
is minimal.
The converse of Proposition 3.14 is not true. If, for instance, B =
C([0, 1] ∪ [2, 3]), then the function which is 1 on [0, 1] and 0 on [2, 3] is a
minimal projection in B which is not of rank one. We will see now that for
the algebras we shall be concerned with, the classes of minimal projections
and of rank one projections coincide.
Theorem 3.15. Let A be a unital fractal C ∗ -subalgebra of F which contains the ideal G. Then every minimal projection in (A ∩ K)/G is a rank one
projection in A/G.
Algebras of approximation sequences
67
Proof. Let q be a minimal projection in (A ∩ K)/G. If q = 0, then q is
of rank one. So let q 6= 0 and a ∈ A/G. We have to show that there is an
α ∈ C such that qaq = αq. Since a can be written as a linear combination of
four non-negative elements, it is sufficient to verify that, for every b ∈ A/G,
there is a β ∈ C such that
qb∗ bq = βq.
(13)
Let C stand for the smallest C ∗ -subalgebra of A/G which contains the elements q and qb∗ bq. This algebra is commutative, has q as its identity element,
and its maximal ideal space is homeomorphic to σC (qb∗ bq).
Choose sequences (Qn ) ∈ A ∩ K and (Bn ) ∈ A with (Qn ) + G = q and
(Bn ) + G = b. From Corollary 3.9 we infer that
σA ((Qn Bn∗ Bn Qn )n≥1 ) = {0} ∪ {λ1 , λ2 , . . .} =: Λ
with numbers λ1 > λ2 > . . . > 0 having 0 as their only possible accumulation
point. Hence, σA/G (qb∗ bq) ⊆ Λ. We claim that
σC (qb∗ bq) ⊆ σA/G (qb∗ bq),
(14)
which clearly implies that
σC (qb∗ bq) ⊆ Λ.
(15)
To prove (14), let λ ∈ C\σA/G (qb∗ bq). Then there is a c in A/G such that
(qb∗ bq − λe)c = e. Multiplying this identity by q we get (qb∗ bq − λq)qcq = q,
i.e. qb∗ bq − λq is invertible in the C ∗ -algebra q(A/G)q with identity q. But
then, by the inverse closedness of C ∗ -algebras, qb∗ bq − λq is also invertible in
the C ∗ -subalgebra C of q(A/G)q. Hence, λ ∈ C\σC (qb∗ bq), which yields (14).
Now assume that σC (qb∗ bq) contains at least two points. Then at least
one of them, say λ0 , is not 0. We consider the element p ∈ C whose Gelfand
transform satisfies
p(λ0 ) = 1,
p(λ) = 0 for all λ ∈ σC (qb∗ bq) \ {λ0 }
(this function is continuous on σC (qb∗ bq) due to (15) and the choice of λ0 ).
The element p is a projection in C ⊆ (A ∩ K)/G, and p q since q(λ) = 1 for
all λ ∈ σC (qb∗ bq). But, evidently, p is neither 0 (since p(λ0 ) = 1) nor q (since
σC (qb∗ bq) contains at least two points by assumption). This contradicts the
minimality of q. Hence, the maximal ideal space σC (qb∗ bq) of C consists of
exactly one point. But then every element of C is a multiple of the identity
element whence there exists a β ∈ C such that (13) holds.
It is also clear that β ≥ 0: Since σA/G (q) = {0, 1}, one has σA/G (βq) =
{0, β} and, consequently,
σA/G (qb∗ bq) = {0, β}.
Because qb∗ bq ≥ 0, this implies β ≥ 0.
68
S. Roch
Minimal generators. The next goal is to show that the ideal (A ∩ K)/G
not only contains minimal projections, but is even generated by its minimal
projections. We start by considering the consequences of the relation r q
between two projections q, r ∈ (A ∩ K)/G for the representatives of these
cosets.
Theorem 3.16. Let A be as in the preceding theorem, and let q, r ∈
(A ∩ K)/G be projections with r q. Let further q = (Qn ) + G and r =
(Rn ) + G with projections (Qn ), (Rn ) ∈ A ∩ K (these projections exist and
can be chosen as essential projections due to Corollary 3.12). If r 6= q, then
dim Im Rn < dim Im Qn
for all n ∈ N with at most finitely many exceptions.
In the proof, we will make use of the following simple observation: Let
P be an orthogonal projection on a Hilbert space H, and let A ∈ L(H) be
an operator such that kP − Ak < 1. Then kP − P AP k = kP (P − A)P k ≤
kP − Ak < 1. Hence, by Neumann series, the operator
P AP = P − (P − P AP ) : Im P → Im P
is invertible, whence
(16)
dim Im P AP = dim Im P.
Proof of Theorem 3.16. Since rqr = r, one has r = rq = qr = rqr.
Consequently,
(17)
(18)
kRn − Qn Rn Qn k → 0,
kRn − Rn Qn k → 0 and kRn − Qn Rn k → 0
Due to (17),
kRn − Qn Rn Qn k < 1
as n → ∞.
for all sufficiently large n.
Thus, by (16),
dim Im Rn = dim Im Rn Qn Rn ≤ dim Im Qn
for all sufficiently large n. Suppose that dim Im Rn = dim Im Qn for all n in
an infinite subset η of N. Then, for every n ∈ η, there is a unitary matrix
Un : Cn → Cn such that Rn = Un Qn Un∗ . (Indeed, choose orthonormal bases
{e1 , . . . , ek } and {f1 , . . . , fk } of Im Qn and Im Rn , complete these bases to
orthonormal bases {e1 , . . . , en } and {f1 , . . . , fn } of Cn , and set Un er := fr
for 1 ≤ r ≤ n.) Thus, for every n ∈ η,
kRn − Rn Qn k = kUn Qn Un∗ − Un Qn Un∗ Qn k = kQn − Qn Un∗ Qn Un k
= kQn − Qn Rn k.
Together with the first assertion of (18), this shows that (Qn − Qn Rn )n∈η
∈ Gη and, together with the second assertion of (18), we even deduce that
Algebras of approximation sequences
69
(Rn − Qn )n∈η ∈ Gη . Via Theorem 3.2, the fractality of A implies that
(Rn − Qn )n∈N ∈ G, i.e. q = r contrary to the hypotheses.
Corollary 3.17. Let A be as in the preceding theorem. Then every
chain q0 q1 q2 . . . of projections qi ∈ (A ∩ K)/G is finite.
Proof. Let (Qn ) ∈ A∩K be a projection such that (Qn )+G = q0 . By the
definition of K, there is a sequence (Kn ) ∈ F with l := supn dim Im Rn < ∞
such that kQn − Rn k < 1 for all n. Then, by (16),
dim Im Qn = dim Im Qn Rn Qn ≤ dim Im Rn ≤ l
(j)
for all n. In particular, l0 := supn dim Im Qn < ∞. For j ≥ 1, let (Qn ) ∈
(j)
A ∩ K be a projection such that (Qn ) + G = qj . Repeated application of
Theorem 3.16 shows that
dim Im Q(1)
for almost all n,
n ≤ l0 − 1
dim Im Q(2)
n ≤ l0 − 2
for almost all n, . . .
where for almost all n means for all n with finitely many exceptions. Thus,
(1)
after at most l0 steps, dim Im Qn = 0 for almost all n, i.e. ql0 = 0.
Corollary 3.18. For every non-trivial projection q ∈ (A ∩ K)/G, there
is a non-trivial minimal projection r ∈ (A ∩ K)/G such that r q.
Proof. If q is not minimal, then there is a projection q1 ∈ (A∩K)/G with
q1 ≺ q. If also q1 is not minimal, then there is a projection q2 ∈ (A ∩ K)/G
such that q2 ≺ q1 , etc. By Corollary 3.17, the chain q q1 q2 . . . has a
smallest non-trivial entry which is a minimal projection.
Corollary 3.19. Every projection q ∈ (A ∩ K)/G is a finite sum of
minimal projections in (A ∩ K)/G.
Proof. If q = 0 then q is minimal. So let q ∈ (A ∩ K)/G be a non-trivial
projection. If q is minimal, then again there is nothing to prove. If q is
not minimal, then there exists a non-trivial minimal projection p1 ≺ q by
Corollary 3.18. The element q − p1 is a projection by Lemma 3.13. If q − p1
is minimal, then q = p1 + (q − p1 ) is a sum of two minimal projections, and
we are done. If q − p1 is not minimal, then (again by Corollary 3.18) there
is a non-trivial minimal projection p2 ≺ q − p1 . If q − p1 − p2 is minimal,
then q = p1 + p2 + (q − p1 − p2 ) is a sum of three minimal projections.
Otherwise, we proceed in this manner to find a non-trivial projection p3
such that p3 ≺ q − p1 − p2 . Since the chain
q q − p 1 q − p 1 − p2 . . .
is finite by Corollary 3.17, this process terminates, i.e. there is a non-trivial
minimal projection q −p1 −. . .−pr . Then q = p1 +. . .+pr +(q −p1 −. . .−pr )
is the sum of r + 1 non-trivial minimal projections.
70
S. Roch
Combining this result with Theorem 3.10 we get:
Theorem 3.20. Let A be a unital fractal C ∗ -subalgebra of F which contains the ideal G. Then the ideal (A∩K)/G is (as an ideal of A/G) generated
by its non-trivial minimal projections.
Equivalently, (A ∩ K)/G is (as an ideal of A/G) generated by its nontrivial projections of rank one (Proposition 3.14 and Theorem 3.15).
Structure of (A ∩ K)/G. To get a complete description of the structure
of the ideal (A ∩ K)/G in case A is a fractal algebra, we need the following
result proved in [5], Theorem 4.
Theorem 3.21. Let B be a unital C ∗ -algebra and k a non-zero rank one
element of B. Further let Id(k) denote the smallest closed ideal of B which
contains k. Then there exists an irreducible representation (H, π) of B such
that
π(Id(k)) = K(H) and Ker(π|Id (k)) = {0}.
Consequently, if k1 and k2 are non-trivial rank one elements in a C ∗ algebra B, then
(19)
either
Id(k1 ) = Id(k2 )
or
Id(k1 ) ∩ Id(k2 ) = {0}.
This alternative suggests introducing an equivalence relation ∼ in the set of
all non-trivial rank one elements in (A ∩ K)/G as follows:
k1 ∼ k 2
if
Id(k1 ) = Id(k2 ).
Let T be the set of equivalence classes of this relation and, for each t ∈ T ,
let It stand for the smallest closed ideal of A/G which is generated by a
non-trivial rank one element k ∈ t.
Theorem 3.22. Let A be a unital fractal C ∗ -subalgebra of F which contains the ideal G.
(a) Each ideal It is ∗ -isometric to the ideal K(Ht ) of all compact operators on a Hilbert space Ht .
(b) If t1 , . . . , tm ∈ T and ti 6= tj for i 6= j, then (It1 + . . . + Itm−1 ) ∩ Itm
= {0}.
(c) (A ∩ K)/G is the smallest closed ideal of A/G which contains every
ideal It .
(d) There is a bijection between T and the spectrum of (A ∩ K)/G.
Proof. Assertion (a) follows from Theorem 3.21, assertion (b) is proved
in [5], and (c) is a consequence of (a) and of Theorem 3.20.
Now let (H, W ) be an irreducible representation of (A ∩ K)/G. Then
there exists a non-trivial minimal projection p ∈ (A ∩ K)/G such that W
does not vanish on Id(p) (otherwise W would be the zero representation due
to (c)). Hence, W |Id(p) is an irreducible representation of Id(p). Suppose
Algebras of approximation sequences
71
that W (q) 6= 0 for another non-trivial minimal projection q 6∼ p. Then
Id(p) ∩ Id(q) = {0}, i.e. qk = kq = 0 for all k ∈ Id(p). Consequently,
W (k)W (q) = W (kq) = 0 = W (qk) = W (q)W (k),
i.e. Im W (q) is an invariant subspace for W (Id(p)). Since W |Id(p) is an irreducible representation, this implies that either Im W (q) = {0} (which
contradicts our assumption W (q) 6= 0) or Im W (q) = H. The latter is impossible if dim H ≥ 2, since W (q) is an operator of rank one (cf.
[5], Theorem 3). It remains to consider the case when Im W (q) = H and
dim H = 1. Then, necessarily, W (p) = 1 and W (q) = 1, hence W (pq) = 1
in contradiction to pq = 0.
Thus, for every irreducible representation (H, W ) of (A ∩ K)/G, there is
exactly one t ∈ T such that W |It 6= 0. Conversely, since It ∼
= K(H) by (a), it
is evident that each ideal It determines (up to unitary equivalence) exactly
one irreducible representation (H, W ) of (A ∩ K)/G with W |It 6= 0.
We mention one more consequence of the preceding theorem. Its proof
is in [5], Theorems 5 and 6. For t ∈ T , we let Wt denote the (essentially
unique) irreducible representation from It to the ideal K(Ht ); its existence
is guaranteed by assertion (a) of the preceding theorem. The extensions of
Wt onto the algebras A/G and A will again be denoted by Wt for simplicity.
Theorem 3.23 (Lifting theorem). Let A and T be as before.
(a) An element a ∈ A/G is invertible in A/G if and only if the operators
Wt (a) are invertible in L(Ht ) for every t ∈ T and if the coset a + (A ∩ K)/G
is invertible in the quotient algebra (A/G)/((A ∩ K)/G).
(b) Let a ∈ A/G be invertible modulo (A ∩ K)/G. Then all operators
Wt (a) are Fredholm, and there are only finitely many t ∈ T for which Wt (a)
is not invertible.
4. Fredholm theory in fractal algebras. Again let A be a unital
fractal C ∗ -subalgebra of F which contains the ideal G. The aim of this section
is to specify the general Fredholm theory to the case of fractal algebras and
to derive a formula for the α-number of a fractal Fredholm sequence. Our
starting point is the following result on the lifting of minimal projections.
Let p ∈ (A ∩ K)/G be a non-trivial minimal projection. Then there is
an essential projection (Πn ) ∈ A ∩ K which lifts p (Corollary 3.12). We will
see now that the behaviour of the sequence (dim Im Πn ) for large n depends
only on the coset of p with respect to the equivalence relation ∼.
Theorem 4.1. Up to a sequence of integers which tends to zero, the
sequence (dim Im Πn )n≥1 is uniquely defined by the coset of p with respect
to the equivalence relation ∼.
We prepare the proof by two simple lemmata.
72
S. Roch
Lemma 4.2. Let B be a C ∗ -algebra, and let p, q be rank one projections
in B with q ∈ Id(p). Then there is a c ∈ Id(p) such that q = c∗ pc.
Proof. By Theorem 3.21, there is a ∗ -isomorphism W from Id(p) onto
K(H) for some Hilbert space H, and W (p) is a non-zero projection of rank
one on H (see [5], Theorem 3). Thus, there is a y ∈ H with kyk = 1 such
that
W (p)x = hx, yiy for all x ∈ H.
Analogously (recall that Id(p) = Id(q) by (19)), there is a z ∈ H with
kzk = 1 such that
Define C ∈ K(H) by
W (q)x = hx, ziz
Cx := hx, ziy
for all x ∈ H.
for x ∈ H.
Then C ∗ x = hx, yiz and C ∗ W (p)Cx = W (q)x for all x ∈ H, as one easily
checks. Since W is a bijection, there is exactly one c ∈ Id(p) with W (c) = C,
and this element c satisfies q = c∗ pc.
Lemma 4.3. Let P and Q be orthogonal projections on a Hilbert space
H and let A, B ∈ L(H) be operators such that kQ − AP Bk < 1. Then
dim Im Q ≤ dim Im P.
Proof. By (16), dim Im Q = dim Im QAP BQ, whence the assertion.
Proof of Theorem 4.1. What we have to show is: If p ∼ q ∈ (A ∩ K)/G
are non-trivial minimal projections with liftings (Πnp ) and (Πnq ), respectively,
then
dim Im Πnp = dim Im Πnq for all sufficiently large n.
First let (Πn1 ) and (Πn2 ) be liftings of one and the same minimal projection
p. Then Πn1 = Πn2 + Gn with a sequence (Gn ) ∈ G. From (16) we conclude
that dim Im Πn1 = dim Im Πn2 for all n with kGn k < 1.
Let now (Πnp ) and (Πnq ) be liftings of p 6= q. By Lemma 4.2, there is a
c ∈ Id(p) = Id(q) such that q = c∗ pc. Let (Cn ) ∈ A ∩ K be a lifting of c.
Then, clearly,
Πnq = Cn∗ Πnp Cn + Gn
with a sequence (Gn ) ∈ G. For n sufficiently large, kGn k < 1, and for these n
we conclude from the preceding lemma that dim Im Πnq ≤ dim Im Πnp . Now
change the roles of p and q to get the reverse inequality.
Thus, with every t ∈ T = Spec (A ∩ K)/G, we can associate a sequence
(αnt )n≥1 by choosing a non-trivial minimal projection p ∈ It with lifting
(Πnp ) and by setting αnt := dim Im Πnp . Recall that this sequence is uniquely
determined up to a sequence tending to zero.
Algebras of approximation sequences
73
Let now A := (An ) ∈ A be a Fredholm sequence, that is, (An ) is invertible modulo K. Then the sequence A is also invertible modulo A ∩ K, and
from Theorem 3.23(b) we conclude that the operators Wt (A) are Fredholm
for every t ∈ T and that only a finite number of them are not invertible.
Thus, the following sums are actually finite:
X
(20)
αn (A) :=
αnt dim Ker Wt (A).
t∈T
Evidently, (20) determines the sequence (αn (A)) only up to a sequence
tending to zero. The main result of the present paper is the following.
Theorem 4.4. Let A be a unital fractal C ∗ -subalgebra of F which contains the ideal G. If A = (An ) is a Fredholm sequence in A, then
(n)
(21)
lim σαn (A) = 0,
n→∞
(n)
where 0 ≤ σ1
(n)
≤ . . . ≤ σn
(n)
lim inf σαn (A)+1 > 0
n→∞
are the singular values of An .
Let us add a few comments on this result before its proof starts.
• For every Fredholm sequence A = (An ) ∈ A, the number
(22)
α(A) := lim sup αn (A)
n→∞
is finite. Since (αn (A)) is a sequence of non-negative integers, it has an
infinite constant subsequence with entries equal to α(A). This shows that
(23)
(n)
lim inf σα(A) = 0,
n→∞
(n)
lim inf σα(A)+1 > 0,
n→∞
i.e. the number α(A) defined by (22) is exactly the α-number of the Fredholm sequence A as defined in Section 2. In that sense, one can consider
(21) as an essential refinement of (10).
• The appearance of the limit in (21) shows that the singular values of a
fractal Fredholm sequence (An ) have the splitting property, i.e. the singular
values of An are contained in [0, εn ] ∪ [δ, ∞) where εn → 0 as n → ∞
and where δ > 0. Thus, the coset (An ) + G is Moore–Penrose invertible in
F/G, which is in perfect accordance with the Moore–Penrose invertibility
(or, equivalently, with the closedness of the range) of a Fredholm operator
on a Hilbert space.
• Observe that the number of the singular values of An which lie in
[0, εn ] can depend on n (it is just given by the number αn (A) in (21)). It is
not hard to construct fractal algebras with arbitrarily prescribed sequences
(αnt ). Consider, for example, for every c ∈ C and every K ∈ K(l 2 ), the
sequence (An ) with
cPn + Pn KPn
if n is even,
(24)
An :=
cPn + Pn KPn + Rn KRn if n is odd,
74
S. Roch
where Pn , Rn : l2 → l2 are the operators mapping (x1 , x2 , . . .) to
(x1 , . . . , xn , 0, 0, . . .) and
(xn , . . . , x1 , 0, 0, . . .),
respectively. The smallest closed subalgebra A of F which contains all of
these sequences is unital and symmetric, and it contains the ideal G. Further,
the mapping
W : A → L(l2 ), (An ) 7→ s-lim An Pn ,
is a representation of A which maps the sequence (24) to the operator cI +K.
In particular, W is an irreducible representation of A. It is also easy to see
that the invertibility of cI + K implies the stability of the sequence (24).
Since every sequence in A is a sum of a sequence of the form (24) and a
sequence in G, we conclude that a sequence (An ) ∈ A is stable if and only
if W (An ) is invertible. This shows that A is a fractal algebra and that the
quotient algebra A/G is ∗ -isometric to the algebra CI + K(l 2 ).
Let (Πn ) be the sequence of the form (24) where c = 0 and K = P1 .
Then p := (Πn ) + G is a non-trivial minimal projection in (A ∩ K)/G, and
one has
n
1 if n is even,
dim Im Πn :=
2 if n is odd.
• In [7, 8], we exclusively considered algebras A of approximation sequences having the property that every non-trivial minimal projection in
(A ∩ K)/G lifts to a sequence of projections of rank one. Similarly, in [5],
we studied the Fredholm theory for sequences in fractal algebras under the
additional hypothesis that the ideal A ∩ K is generated by sequences of projections of rank one. Thus, in both cases, the numbers αnt are independent of
n and can be chosen to be 1 for all n. Under these assumptions, the formula
(20) reduces to
X
α(A) :=
dim Ker Wt (A),
t∈T
occurring in [7, 5].
The remainder of this section is devoted to the proof of Theorem 4.4.
Once we derive the following proposition, the main steps in the proof of
Theorem 4.4 will be as in [5].
Proposition 4.5. Let q, r ∈ F/G be projections with qr = rq = 0, and
let (Rn ) ∈ F be a lifting of r (i.e. all matrices Rn are orthogonal projections,
and (Rn ) + G = r). Then there exists a projection (Qn ) which lifts q such
that
(25)
Rn Qn = Qn Rn = 0 for all sufficiently large n.
Proof. Let (Q0n ) be a lifting of q (which exists by Corollary 3.12) and
consider the matrices Sn := (In − Rn )Q0n (In − Rn ). The coset modulo G of
Algebras of approximation sequences
75
the sequence (Sn ) is
(26)
(e − r)q(e − r) = q − rq − qr + rqr = q.
Hence ((e − r)q(e − r))2 = (e − r)q(e − r), i.e. kSn − Sn2 k → 0. Choose n0
such that kSn − Sn2 k < 1/4 for all n ≥ n0 . Then, by Proposition 3 of [7],
for all n ≥ n0 there is a Gn which belongs to the C ∗ -algebra generated by
In − Rn and Sn such that Sn + Gn is a projection and
kGn k ≤ 2kSn − Sn2 k.
Define Qn := Q0n if n < n0 and Qn := Sn + Gn if n ≥ n0 . Then, for n ≥ n0 ,
kQn − Q0n k = kSn − Q0n + Gn k
≤ k(In − Rn )Q0n (In − Rn ) − Q0n k + kGn k → 0
because of (26). Hence, (Qn ) is also a lifting of q, and this lifting satisfies
(25):
Rn Qn = Rn (In − Rn )Q0n (In − Rn ) + Rn Gn = 0
(recall that Rn Gn = 0 since Gn ∈ alg(In − Rn , Sn )).
Let now (An ) ∈ A be a Fredholm sequence, i.e. (An ) is invertible modulo the ideal A ∩ K. Then, by Theorem 3.23, the operators Wt ((An )) are
Fredholm for every t ∈ T and only finitely many of them are not invertible.
Let PKer Wt ((An )) denote the orthogonal projection from Ht onto the kernel
of Wt ((An )) (only a finite number of these projections is not zero). Decompose each of these projections into a sum of dim Ker Wt ((An )) orthogonal
projections of rank one:
dim Ker Wt ((An ))
PKer Wt ((An )) =
X
Pi,t
i=1
such that Pi,t Pj,t = Pj,t Pi,t = 0 whenever i 6= j (for example, by choosing
an orthonormal basis in Ker Wt ((An )) and by defining Pi,t as the orthogonal
projection onto the ith element of this basis). Since Wt is an isomorphism
between It and K(Ht ), every projection Pi,t corresponds uniquely to a coset
pi,t ∈ It . Clearly, pi,t pj,t = pj,t pi,t = 0 if i 6= j. Since Is ∩ It = {0} for all
s, t ∈ T with s 6= t, one has moreover
(27)
pi,t pj,s = pj,s pi,t = 0
whenever (i, t) 6= (j, s).
Using Proposition 4.5 it is not hard to see that the cosets pi,t can be lifted
to sequences (Pni,t ) in such a way that
(28)
Pni,t Pnj,s = Pnj,s Pni,t = 0
whenever (i, t) 6= (j, s)
for all sufficiently large n. Indeed, suppose we have already lifted a finite
number of the projections pi,t in such a way that (28) holds. Let (Rn ) be
the sum of these liftings. Then every Rn with n large enough is an orthogonal
76
S. Roch
projection due to (28). If q is an element of the family of the projections
pi,t which has not yet been lifted, then we construct a lifting (Qn ) of q such
that Rn Qn = Qn Rn = 0 for all sufficiently large n, which is possible due
to Proposition 4.5. It is obvious that the new set of all liftings (enlarged
by (Qn )) again satisfies the condition (28). This procedure terminates since
there are only a finite number of projections to lift.
The orthogonality (28) ensures that the operators
Rn :=
Wt ((An ))
X dim KerX
Pni,t
i=1
t∈T
are orthogonal projections for all sufficiently large n and that
(29)
dim Im Rn =
Wt ((An ))
X dim KerX
αnt =
i=1
t∈T
X
αnt dim Ker Wt (A)
t∈T
= αn (A)
with A = (An ) for all sufficiently large n. As in [5] one can now easily check
that
(30)
(An )∗ (An ) + (Rn )
is a stable sequence and
(An )∗ (An )(Rn ) ∈ G.
Thus, the coset (An ) + G is Moore–Penrose invertible in A/G, and (Rn ) + G
is the associated Moore–Penrose projection. It is shown in Theorem 3 of
[7] that there is then a sequence (Πn ) of orthogonal projections such that
kΠn − Rn k → 0 and that every projection Πn belongs to the C ∗ -algebra
generated by A∗n An and by the identity matrix In . From kΠn − Rn k → 0 we
conclude that
dim Im Πn = dim Im Rn = αn (A)
for all sufficiently large n,
and the property Πn ∈ alg(A∗n An , In ) ensures that the matrices A∗n An and
Πn can be diagonalized simultaneously:
(n)
Un∗ A∗n An Un = diag(λ1 , . . . , λ(n)
n ),
(n)
Un∗ Πn Un = diag(π1 , . . . , πn(n) )
with unitary matrices Un which we choose in such a way that
(n)
0 ≤ λ1
≤ . . . ≤ λ(n)
n .
The conditions (30) (with Πn in place of Rn ) imply the existence of a C > 0
such that
(31)
(32)
(n)
λi
(n)
+ πi
(n)
λi
·
≥C>0
(n)
πi
<C
for all n ≥ n0 and i = 1, . . . , n,
for all n ≥ n0 and i = 1, . . . , n,
Algebras of approximation sequences
77
Let i = in be such that
(n)
λ1
(n)
≤ . . . ≤ λi
(n)
< c ≤ λi+1 ≤ . . . ≤ λ(n)
n .
Then, for (31) to hold it is necessary that
(n)
π1
(n)
= . . . = πi
=1
whereas (32) requires that
(n)
πi+1 = . . . = λ(n)
n = 0.
Hence, in = αn (A) and
Un∗ Πn Un = diag(1, . . . , 1, 0, . . . , 0)
with αn (A) ones and n − αn (A) zeros. Now the identity (21) follows from
the diagonalized form of (30) with Πn in place of Rn . This finishes the proof
of Theorem 4.4
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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New York, 2000.
G. K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, Academic Press,
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S. Roch, Algebras of approximation sequences: Fractality, in: Problems and Methods
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S. Roch, Algebras of approximation sequences: Fredholmness, preprint 2048(1999) TU
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—, —, Index calculus for approximation methods and singular value decomposition,
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Fachbereich Mathematik
Technische Universität Darmstadt
Schlossgartenstrasse 7
64289 Darmstadt, Germany
E-mail: [email protected]
Received 27 December 2000
(4658)